173 



(where '/^m represents the mean vahie of V', (the moment of 

 momentum of the electron). 

 It will be assumed : 



1. that in A„(pi p^ p^) all the /; occur, in sucli a way that between 

 tlie three differential quotients ^^^"/d/'i ^^'^"/dj,^, ^'^"/d;-» there do not 

 exist any rational relations ; 



2. that 7 is very large, and that the quantity '^'a// is small as 

 compared to the mean angular velocity of the electron, so thai the 

 second and higher powers of this quantity may be neglected. 



It is then very easy to tind solutions of the problem considered, 

 following a method given by Delaunay and Whittaker ^) ; if these 

 solutions are restricted to the terms which contain "^V/ ^o ^'^e powers 

 and 1, they are of the form: 





xn 



.. = a. + y.^ 



cos 



(6) 



( Sin \ 



Here Q^ Q^ Q^ are new angular variables; P^ P^ P^ are the 

 canonical momenta, corresponding to them. 



The total energy of the system is found to be (to the same 

 degree of approximation) : 



a=A, (P^P^P^)-^. y>;„ (AP,P.) + If ') . . • (7) 



§ 3. The Quantum formulae. 



Following the ideas developed by Schwarzschild '), the quantum 

 formulae for the system may be introduced as follows : 



the quantities P^ P^ P^ ^\ are put equal to integral multiples of 

 h 



h h h h 



The energy, when expressed as a function of the quantum 

 numbers n^n^n^n^^ becomes: 



a = a, (n,n,n,) - n, . ^-j— -^ n, ^- ... (J) 



1) Gf. Whittaker, 1. c. p. 404. 



2) The three terms of this equation may be interpreted approximately as follows: 

 Af, is the energy of the electron ; ^j-'^i is the energy of rotation of the molecule; 



the term — '- — " is related to the GoRiOLis-reaclion generated by the rotation. 



^) K. Schwarzschild, 1. c. 



